\subsubsection{Comb Filter}

A comb filter is a feedback filter that has the following equation:

$y_{t}=x_{t}+R^Ly_{t-L}$

where $y_{t}$ is the output and $x_{t}$ is the input.  A comb filter is produced when a signal is first delayed and then added back to itself. The delayed signal can also be different from the original, the only requirement is that both signals have similar frequency content. The result of the difference in phase of the waveforms, is some very deep notches occurring regularly in the signal, causing some frequencies to disappear, while the frequencies between these notches are boosted by a certain amount of db. An example of this effect is shown in figure \ref{fig:7samples}, where a signal consisting of white noise, running on the L and R channel, is illustrated in a linear spectrogram. The R channel is delayed by 7 samples (converted to milliseconds), and then combined back with the L channel. As comparison, figure \ref{fig:original} is an illustration of how a normal white noise signal looks in a spectrogram.\cite{combfilter_thestudio}

\begin{figure}[htbp] \centering
\begin{minipage}[b]{0.45\textwidth} \centering
\includegraphics[width=0.97\textwidth]{images/CombFilter/combFil1} % Venstre billede
\end{minipage} \hfill
\begin{minipage}[b]{0.45\textwidth} \centering
\includegraphics[width=0.97\textwidth]{images/CombFilter/combFil2} % Højre billede
\end{minipage} \\ % Captions og labels
\begin{minipage}[t]{0.45\textwidth}
\caption{Comb Filter with 7 samples delay.} % Venstre caption og label
\label{fig:7samples}
\end{minipage} \hfill
\begin{minipage}[t]{0.45\textwidth}
\caption{Standard white noise.} % Højre caption og label
\label{fig:original}
\end{minipage}
\end{figure}

As the delay increases, the number of notches increases as well. This is illustrated in figure \ref{fig:17samples}, where the delay is 17 samples long (converted to milliseconds). The comb filter got its name for looking like a comb when plotted on a frequency response graph(logarithmic) as seen in figure \ref{fig:combFilter}.\cite{combfilter_thestudio}

\begin{figure}[htbp] \centering
\begin{minipage}[b]{0.45\textwidth} \centering
\includegraphics[width=0.97\textwidth]{images/CombFilter/combFil3} % Venstre billede
\end{minipage} \hfill
\begin{minipage}[b]{0.45\textwidth} \centering
\includegraphics[width=0.90\textwidth]{images/CombFilter/combFil4} % Højre billede
\end{minipage} \\ % Captions og labels
\begin{minipage}[t]{0.45\textwidth}
\caption{Comb Filter with 17 samples delay.} % Venstre caption og label
\label{fig:17samples}
\end{minipage} \hfill
\begin{minipage}[t]{0.45\textwidth}
\caption{Comb filter on a frequency response graph.} % Højre caption og label
\label{fig:combFilter}
\end{minipage}
\end{figure}

The comb filter often appears if an object is recorded by multiple microphones, which are placed at different distances from the object. This result is two signals of which the one is delayed, simply because the sound will reach the closest microphone first and then the microphone that is further away from the object afterwards.\cite{combfilter_thestudio} The sound this creates is very often like a metallic resonance, which can dramatically change the tonal color of the signal.\cite{combfilter_sweet}